Relatório de Análise de Séries Temporais
##
## Attaching package: 'zoo'
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## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric
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## This is forecast 5.6
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## locfit 1.5-9.1 2013-03-22
## Loading required package: nlme
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## Attaching package: 'nlme'
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## The following object is masked from 'package:forecast':
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## getResponse
##
## This is mgcv 1.8-3. For overview type 'help("mgcv-package")'.Este relatório tem como objecto de análise
Dados originais:
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov
## 1956 1709 1646 1794 1878 2173 2321 2468 2416 2184 2121 1962
## 1957 1751 1688 1920 1941 2311 2279 2638 2448 2279 2163 1941
## 1958 1773 1688 1783 1984 2290 2511 2712 2522 2342 2195 1931
## 1959 1730 1688 1899 1994 2342 2553 2712 2627 2363 2311 2026
## 1960 1762 1815 2005 2089 2617 2828 2965 2891 2532 2363 2216
## 1961 1804 1773 2015 2089 2627 2712 3007 2880 2490 2237 2205
## 1962 1868 1815 2047 2142 2743 2775 3028 2965 2501 2501 2131
## 1963 1910 1868 2121 2268 2690 2933 3218 3028 2659 2406 2258
## 1964 1889 1984 2110 2311 2785 3039 3229 3070 2659 2543 2237
## 1965 1962 1910 2216 2437 2817 3123 3345 3112 2659 2469 2332
## 1966 1910 1941 2216 2342 2923 3229 3513 3355 2849 2680 2395
## 1967 1994 1952 2290 2395 2965 3239 3608 3524 3018 2648 2363
## 1968 1994 1941 2258 2332 3323 3608 3957 3672 3155 2933 2585
## 1969 2057 2100 2458 2638 3292 3724 4652 4379 4231 3756 3429
## 1970 3345 4220 4874 5064 5951 6774 7997 7523 7438 6879 6489
## 1971 5919 6183 6594 6489 8040 9715 9714 9756 8595 7861 7753
## 1972 7778 7402 8903 9742 11372 12741 13733 13691 12239 12502 11241
## 1973 11569 10397 12493 11962 13974 14945 16805 16587 14225 14157 13016
## 1974 11704 12275 13695 14082 16555 17339 17777 17592 16194 15336 14208
## 1975 12354 12682 14141 14989 16159 18276 19157 18737 17109 17094 15418
## 1976 13260 14990 15975 16770 19819 20983 22001 22337 20750 19969 17293
## 1977 15117 16058 18137 18471 21398 23854 26025 25479 22804 19619 19627
## 1978 17243 18284 20226 20903 23768 26323 28038 26776 22886 22813 22404
## 1979 18839 18892 20823 22212 25076 26884 30611 30228 26762 25885 23328
## 1980 21433 22369 24503 25905 30605 34984 37060 34502 31793 29275 28305
## 1981 27730 27424 32684 31366 37459 41060 43558 42398 33827 34962 33480
## 1982 30715 30400 31451 31306 40592 44133 47387 41310 37913 34355 34607
## 1983 26138 30745 35018 34549 40980 42869 45022 40387 38180 38608 35308
## 1984 28801 33034 35294 33181 40797 42355 46098 42430 41851 39331 37328
## 1985 32494 33308 36805 34221 41020 44350 46173 44435 40943 39269 35901
## 1986 31239 32261 34951 38109 43168 45547 49568 45387 41805 41281 36068
## 1987 32791 34206 39128 40249 43519 46137 56709 52306 49397 45500 39857
## 1988 35567 37696 42319 39137 47062 50610 54457 54435 48516 43225 42155
## 1989 37541 37277 41778 41666 49616 57793 61884 62400 50820 51116 45731
## 1990 40459 40295 44147 42697 52561 56572 56858 58363 45627 45622 41304
## 1991 35592 35677 39864 41761 50380 49129 55066 55671 49058 44503 42145
## 1992 38963 38690 39792 42545 50145 58164 59035 59408 55988 47321 42269
## 1993 37059 37963 31043 41712 50366 56977 56807 54634 51367 48073 46251
## 1994 39975 40478 46895 46147 55011 57799 62450 63896 57784 53231 50354
## 1995 41600 41471 46287 49013 56624 61739 66600 60054
## Dec
## 1956 1825
## 1957 1878
## 1958 1910
## 1959 1910
## 1960 2026
## 1961 1984
## 1962 2015
## 1963 2057
## 1964 2142
## 1965 2110
## 1966 2205
## 1967 2247
## 1968 2384
## 1969 3461
## 1970 6288
## 1971 8154
## 1972 10829
## 1973 12253
## 1974 13116
## 1975 14312
## 1976 16498
## 1977 18488
## 1978 19795
## 1979 21930
## 1980 25248
## 1981 32445
## 1982 28729
## 1983 30234
## 1984 34514
## 1985 32142
## 1986 34879
## 1987 37958
## 1988 39995
## 1989 42528
## 1990 36016
## 1991 38698
## 1992 39606
## 1993 43736
## 1994 38410
## 1995
## Warning: p-value smaller than printed p-value
## Warning: p-value smaller than printed p-value
## Warning: p-value smaller than printed p-value
## Warning: p-value smaller than printed p-value
Análise de Sazonalidade
Sazonalidade - Autocorrelações
Seasonal Decomposition of Time Series by Loess
Outros testes
McLeod-Li test for conditional heteroscedascity (ARCH)
Projeções
Space-State Model (ETS)
## ETS(M,Md,M)
##
## Call:
## ets(y = d)
##
## Smoothing parameters:
## alpha = 0.6534
## beta = 0.0282
## gamma = 1e-04
## phi = 0.98
##
## Initial states:
## l = 2085.0093
## b = 0.9965
## s=0.8476 0.919 1.004 1.072 1.206 1.264
## 1.176 1.093 0.9198 0.8936 0.8085 0.7973
##
## sigma: 0.0486
##
## AIC AICc BIC
## 8991 8992 9062
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 10.56 1454 791.2 0.1444 3.536 0.4582 -0.1029##
## Forecast method: ETS(M,Md,M)
##
## Model Information:
## ETS(M,Md,M)
##
## Call:
## ets(y = d)
##
## Smoothing parameters:
## alpha = 0.6534
## beta = 0.0282
## gamma = 1e-04
## phi = 0.98
##
## Initial states:
## l = 2085.0093
## b = 0.9965
## s=0.8476 0.919 1.004 1.072 1.206 1.264
## 1.176 1.093 0.9198 0.8936 0.8085 0.7973
##
## sigma: 0.0486
##
## AIC AICc BIC
## 8991 8992 9062
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 10.56 1454 791.2 0.1444 3.536 0.4241 -0.1029
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 1995 54570 51137 57878 49256 59660
## Oct 1995 51150 47349 55056 45314 57023
## Nov 1995 46882 42894 50972 40954 53380
## Dec 1995 43284 39087 47544 37140 50128
## Jan 1996 40760 36422 45294 34342 47917
## Feb 1996 41378 36568 46515 34045 49494
## Mar 1996 45775 40087 51828 37171 55471
## Apr 1996 47167 40710 54053 37507 58303
## May 1996 56089 47815 64670 43979 70183
## Jun 1996 60426 51093 70289 47329 77135
## Jul 1996 64992 54329 76487 49810 83908
## Aug 1996 62055 51451 73892 46707 81565
## Sep 1996 55226 45372 66325 40804 74255
## Oct 1996 51753 42195 62425 37902 69898
## Nov 1996 47424 38220 57847 34245 64722
## Dec 1996 43774 34913 54193 30991 60944
## Jan 1997 41212 32500 51249 28656 58211
## Feb 1997 41827 32605 52446 28595 59800
## Mar 1997 46263 35796 58630 31316 67271
## Apr 1997 47659 36690 61041 31857 70797
## May 1997 56663 42965 72758 37049 84628
## Jun 1997 61032 45728 79314 39068 93416
## Jul 1997 65630 48831 85912 41708 101327
## Aug 1997 62652 45978 82976 39205 97444
## Sep 1997 55747 40502 74584 34359 88741
## Oct 1997 52232 37598 70399 31669 83769
## Nov 1997 47853 34132 65100 28402 77776
## Dec 1997 44163 31169 60643 26054 72734
## Jan 1998 41571 29133 57514 23942 69832
## Feb 1998 42184 29200 58911 24022 71336
## Mar 1998 46649 31985 65625 25948 79938
## Apr 1998 48049 32716 68266 26541 83150
## May 1998 57117 38359 81873 31053 100488
## Jun 1998 61511 40801 89073 32780 111315
## Jul 1998 66135 43719 96478 34858 121885
## Aug 1998 63125 41010 92952 32924 117214
## Sep 1998 56159 36372 83407 28639 104750
## Oct 1998 52610 33532 78594 26581 100785
## Nov 1998 48193 30408 72846 23877 92543
## Dec 1998 44470 27816 67782 21716 87136
## Jan 1999 41854 25932 64425 20192 82708
## Feb 1999 42466 26037 66118 20023 84462
## Mar 1999 46954 28437 73605 21964 95385
## Apr 1999 48358 29159 76553 22560 99166
## May 1999 57476 34310 91106 26256 119301
## Jun 1999 61890 36358 99770 27951 130750
## Jul 1999 66535 38999 107877 29684 142448
## Aug 1999 63499 36808 104178 27890 137450
## Sep 1999 56485 32650 93215 24438 123808
## Oct 1999 52909 30180 88217 22831 117597
## Nov 1999 48461 27432 81644 20371 110571
## Dec 1999 44713 25115 76383 18614 103450
Arima - Pr? an?lise
Arima - Autocorrelation
Arima - Partial Autocorrelation
Arima - Proje??o
##
## ARIMA(2,1,2)(1,0,1)[12] with drift : Inf
## ARIMA(0,1,0) with drift : 8882
## ARIMA(1,1,0)(1,0,0)[12] with drift : 8506
## ARIMA(0,1,1)(0,0,1)[12] with drift : 8695
## ARIMA(1,1,0) with drift : 8841
## ARIMA(1,1,0)(2,0,0)[12] with drift : 8456
## ARIMA(1,1,0)(2,0,1)[12] with drift : Inf
## ARIMA(0,1,0)(2,0,0)[12] with drift : 8523
## ARIMA(2,1,0)(2,0,0)[12] with drift : 8450
## ARIMA(2,1,1)(2,0,0)[12] with drift : 8422
## ARIMA(3,1,2)(2,0,0)[12] with drift : Inf
## ARIMA(2,1,1)(2,0,0)[12] : 8420
## ARIMA(2,1,1)(1,0,0)[12] : 8467
## ARIMA(2,1,1)(2,0,1)[12] : Inf
## ARIMA(1,1,1)(2,0,0)[12] : Inf
## ARIMA(3,1,1)(2,0,0)[12] : Inf
## ARIMA(2,1,0)(2,0,0)[12] : 8448
## ARIMA(2,1,2)(2,0,0)[12] : 8421
## ARIMA(1,1,0)(2,0,0)[12] : 8454
## ARIMA(3,1,2)(2,0,0)[12] : Inf
##
## Best model: ARIMA(2,1,1)(2,0,0)[12]##
## Forecast method: ARIMA(2,1,1)(2,0,0)[12]
##
## Model Information:
## Series: d
## ARIMA(2,1,1)(2,0,0)[12]
##
## Coefficients:
## ar1 ar2 ma1 sar1 sar2
## 0.472 0.219 -0.956 0.554 0.352
## s.e. 0.061 0.058 0.035 0.046 0.046
##
## sigma^2 estimated as 2807497: log likelihood=-4210
## AIC=8431 AICc=8432 BIC=8456
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 100.8 1674 955.2 0.7066 4.236 0.512 -0.01898
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 1995 57436 55288 59583 54152 60720
## Oct 1995 53216 50801 55632 49522 56911
## Nov 1995 51148 48499 53796 47097 55198
## Dec 1995 43604 40823 46385 39351 47857
## Jan 1996 44065 41190 46941 39668 48463
## Feb 1996 44172 41229 47114 39671 48672
## Mar 1996 49107 46114 52100 44529 53685
## Apr 1996 50357 47324 53389 45718 54995
## May 1996 57701 54636 60766 53013 62389
## Jun 1996 61520 58427 64613 56790 66250
## Jul 1996 65855 62738 68971 61088 70621
## Aug 1996 62737 59599 65875 57938 67536
## Sep 1996 59132 55637 62628 53786 64479
## Oct 1996 55190 51567 58813 49649 60731
## Nov 1996 53029 49288 56771 47308 58751
## Dec 1996 44639 40814 48463 38790 50488
## Jan 1997 46019 42127 49912 40066 51972
## Feb 1997 46033 42085 49981 39995 52071
## Mar 1997 50466 46470 54462 44354 56577
## Apr 1997 52119 48081 56157 45943 58295
## May 1997 58872 54796 62949 52639 65106
## Jun 1997 62792 58681 66903 56505 69079
## Jul 1997 66908 62764 71051 60570 73245
## Aug 1997 62873 58698 67047 56488 69257
## Sep 1997 59952 55359 64545 52927 66977
## Oct 1997 56280 51526 61033 49010 63549
## Nov 1997 54353 49450 59257 46854 61852
## Dec 1997 47044 42031 52057 39377 54711
## Jan 1998 47972 42867 53076 40165 55779
## Feb 1998 48017 42835 53199 40092 55942
## Mar 1998 52213 46964 57462 44185 60241
## Apr 1998 53570 48260 58880 45450 61690
## May 1998 59901 54537 65266 51697 68106
## Jun 1998 63420 58004 68836 55136 71703
## Jul 1998 67229 61764 72694 58871 75587
## Aug 1998 63894 58382 69405 55465 72322
## Sep 1998 61004 55172 66837 52085 69924
## Oct 1998 57579 51600 63558 48435 66724
## Nov 1998 55750 49632 61869 46393 65108
## Dec 1998 48742 42514 54970 39217 58267
## Jan 1999 49742 43420 56065 40072 59413
## Feb 1999 49772 43366 56179 39975 59570
## Mar 1999 53661 47179 60142 43748 63573
## Apr 1999 54995 48444 61546 44976 65014
## May 1999 60885 54269 67501 50767 71003
## Jun 1999 64216 57539 70894 54004 74429
## Jul 1999 67778 61041 74514 57475 78080
## Aug 1999 64507 57714 71301 54117 74897
## Sep 1999 61876 54784 68968 51030 72723
## Oct 1999 58684 51442 65925 47609 69759
## Nov 1999 56991 49606 64376 45697 68285
## Dec 1999 50531 43030 58032 39059 62002
Linear Model
##
## Call:
## lm(formula = d ~ time(d))
##
## Residuals:
## Min 1Q Median 3Q Max
## -16988 -5263 -1184 4830 19858
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.01e+06 5.04e+04 -59.6 <2e-16 ***
## time(d) 1.53e+03 2.55e+01 60.0 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6380 on 474 degrees of freedom
## Multiple R-squared: 0.884, Adjusted R-squared: 0.883
## F-statistic: 3.6e+03 on 1 and 474 DF, p-value: <2e-16
## Analysis of Variance Table
##
## Response: d
## Df Sum Sq Mean Sq F value Pr(>F)
## time(d) 1 1.46e+11 1.46e+11 3603 <2e-16 ***
## Residuals 474 1.93e+10 4.06e+07
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Trend - Seazonal Linear Model
##
## Call:
## lm(formula = formula, data = "d", na.action = na.exclude)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14888 -4144 -586 3949 15668
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -12909.11 994.61 -12.98 < 2e-16 ***
## trend 127.42 1.88 67.77 < 2e-16 ***
## season2 351.28 1260.19 0.28 0.781
## season3 1885.52 1260.19 1.50 0.135
## season4 2211.50 1260.20 1.75 0.080 .
## season5 5704.76 1260.21 4.53 7.6e-06 ***
## season6 7552.62 1260.22 5.99 4.1e-09 ***
## season7 9243.11 1260.24 7.33 1.0e-12 ***
## season8 8164.92 1260.26 6.48 2.4e-10 ***
## season9 5321.77 1268.25 4.20 3.3e-05 ***
## season10 3946.18 1268.25 3.11 0.002 **
## season11 2299.22 1268.26 1.81 0.070 .
## season12 456.53 1268.28 0.36 0.719
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5640 on 463 degrees of freedom
## Multiple R-squared: 0.911, Adjusted R-squared: 0.909
## F-statistic: 396 on 12 and 463 DF, p-value: <2e-16## Analysis of Variance Table
##
## Response: d
## Df Sum Sq Mean Sq F value Pr(>F)
## trend 1 1.46e+11 1.46e+11 4610.9 <2e-16 ***
## season 11 4.56e+09 4.15e+08 13.1 <2e-16 ***
## Residuals 463 1.47e+10 3.18e+07
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Compara??o de Modelos
Erros
Residuos ETS
Residuos ARIMA
Residuos TSLM
